Premium
Ramsey, Paper, Scissors
Author(s) -
Fox Jacob,
He Xiaoyu,
Wigderson Yuval
Publication year - 2020
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20950
Subject(s) - graph , combinatorics , diagonal , mathematics , ramsey's theorem , discrete mathematics , mathematical economics , geometry
We introduce a graph Ramsey game called Ramsey, Paper, Scissors. This game has two players, Proposer and Decider. Starting from an empty graph on n vertices, on each turn Proposer proposes a potential edge and Decider simultaneously decides (without knowing Proposer's choice) whether to add it to the graph. Proposer cannot propose an edge which would create a triangle in the graph. The game ends when Proposer has no legal moves remaining, and Proposer wins if the final graph has independence number at least s . We prove a threshold phenomenon exists for this game by exhibiting randomized strategies for both players that are optimal up to constants. Namely, there exist constants 0 < A < B such that (under optimal play) Proposer wins with high probability if s < A n log n , while Decider wins with high probability if s > B n log n . This is a factor of Θ ( log n ) ) larger than the lower bound coming from the off‐diagonal Ramsey number r (3, s ).